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" MAGNITUDES which have the same ratio to the same magnitude are equal to one another ; and those to which the same magnitude has the same ratio are equal to one another. "
The Solutions of the Geometrical Problems: Consisting Chiefly of Examples in ... - Page 16
by Thomas Gaskin - 1847 - 263 pages
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Euclid's Elements of geometry, transl. To which are added, algebraic ...

Euclides - 1826 - 226 pages
...be taken greater than p). It is 5. evident that ^* > - f , and £ < i. QED PROPOSITION IX. THEOREM. Magnitudes which have the same ratio to the same magnitude,...magnitude has the same ratio are equal to one another. the same ratio« to c, but they have: therefore • 8.5. Again, let c have the same ratio to A and...
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An Elementary Treatise on Algebra: Theoretical and Practical ...

James Ryan - Algebra - 1826 - 430 pages
...quotient ; and therefore the ratio of C to B is greater than the ratio of C i& AQED PROP. IX. TIIEOR. Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio, are equal to one another. DEMONSTRATION. 1 . Let A and B have the same ratio to C, it is to be proved that A is equal to B. Because...
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Elements of Geometry, Containing the First Six Books of Euclid

Euclid - 1826 - 234 pages
...Def.?. 5. evident that -* > -, and ? < £. QED ppmn PROPOSITION IX. THEOREM. Magnitudes which hate the same ratio to the same magnitude, are equal to...magnitude has the same ratio are equal to one another. For let each of the magnitudes A, B, have the same ratio to c : then is A equal to B. 129 the same...
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Elements of Geometry: Containing the First Six Books of Euclid, with a ...

Euclid, John Playfair - Euclid's Elements - 1826 - 326 pages
...ratio to A than it has to A+B [def. 7. 5.). Therefore, &e. QE JL». PROP. IX. THEOR. Magnitudes whieh have the same ratio to the same magnitude are equal to one another ; and those to whieh the same magnitude has the same ratio are equal to one another. If A. : C ". B : C, A=B. For,...
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The Elements of Euclid: The Errors by which Theon, Or Others, Have Long ...

Robert Simson - Trigonometry - 1827 - 546 pages
...magnitudes, &c. QED PROP. IX. THEOR. Magnitudes which have the same ratio to the same mag- See N. nitude are equal to one another : and those to which the...magnitude has the same ratio are equal to one another. Let A, B have each of them the same ratio to C: A shall be equal to B. For, if they are not equal,...
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The First Six Books of the Elements of Euclid, with a Commentary and ...

Euclid, Dionysius Lardner - Euclid's Elements - 1828 - 542 pages
...A, but not greater than m (A + B), C : A is greater than C : A + B. PROPOSITION IX. THEOREM. (486) Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio are equal to one another. I,et A, B have each of them the same ratio to C : A is equal toB. For, if they are not equal, one of...
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Geometry, Plane, Solid, and Spherical, in Six Books: To which is Added, in ...

Pierce Morton - Geometry - 1830 - 584 pages
...or a greater ratio to В than it has to A. (See also 9. Cor. 3.) Therefore, &c. Cor. 1. (Eue. v. 9.) Magnitudes which have the same ratio to the same magnitude are equal tu one another : as likewise those to which the same magnitude has the same ratio. Cor. 2. A ratio...
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The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ...

Euclid - 1835 - 540 pages
...Wherefore, " of unequal magnitudes," &c. QED 0 B K A C GB LKHD b 7. def. 5. PROP. IX. THEOR. D See N. Magnitudes which have the same ratio to the same magnitude...magnitude has the same ratio are equal to one another. Let A, B, have each of them the same ratio to C ; then will A be equal to B : For, if they are not...
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Library of Useful Knowledge: Geometry plane, solid, and spherical [by Pierce ...

Mathematics - 1835 - 684 pages
...a greater ratio to В than it has to .\. (See also У. Cor. 3.) Therefore, &c. Cor. I. (Euc.v. 9.) Magnitudes which have the same ratio to the same magnitude are equal to one another : as likewise those to which the same magnitude has the same ratio. Cor. 2. A ratio which is compounded...
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The Elements of Euclid: viz. the first six books, together with the eleventh ...

Robert Simson - Trigonometry - 1835 - 544 pages
...has to AB. Wherefore, " of unequal magnitudes," &c. QED C B HD b7. def. 5. PROP. IX. THEOR. D See N Magnitudes which have the same ratio to the same magnitude are equal to one anotlier ; and those to which the same magnitude has the same ratio are equal to one another. Let A,...
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